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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] of dimension $n \\ge 3$, with [[Levi-Civita connection|CAZYGE1T4E]] $\\nabla$, [[Ricci curvature|4GTNM0WHDH]] $\\operatorname{Ric}$ and [[scalar curvature|ZHX2VYQDZ3]] $S$.\nLet $P = \\frac{1}{n - 2} \\left( \\operatorname{Ric} - \\frac{S}{2(n - 1)} g \\right)$ be the Schouten tensor.\nThe *Cotton tensor* of $(M, g)$ is the $(0,3)$-tensor\n$$C(X, Y, Z) = (\\nabla_Z P)(X, Y) - (\\nabla_Y P)(X, Z),$$\nfor vector fields $X, Y, Z$; in index notation $C_{ijk} = \\nabla_k P_{ij} - \\nabla_j P_{ik}$.\n\n## Remarks\n\nExample: if $(M, g)$ is Einstein with $\\operatorname{Ric} = \\lambda g$ for a constant $\\lambda$ and $n \\ge 3$, then $S = n\\lambda$ is constant and $P$ is a constant multiple of $g$.\nSince $\\nabla g = 0$, also $\\nabla P = 0$, so $C = 0$.\n\nNon-example: for $n = 2$ the factor $1/(n - 2)$ is not defined, so there is no Cotton tensor of a surface in this sense.\n\n$C$ is antisymmetric in its last two arguments, and its trace over the first two arguments vanishes by the contracted Bianchi identity.\nFor $n = 3$, $C = 0$ if and only if $g$ is locally conformally flat.\nFor $n \\ge 4$, the divergence of the [[Weyl tensor|SBHY31HESP]] is $(n - 3)$ times $C$, up to a sign fixed by the conventions.\nSome texts use the opposite sign or a different order of indices.\n",
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